Definition:Principal Square Root/Also presented as

Principal Square Root: Also presented as

Let $z \in \C$ be a complex number.

The principal square root of $z = x + i y$ can also be seen presented in the form:

$z^{1/2} = \paren {\dfrac 1 2 \paren {r + x} }^{1/2} \pm i \paren {\dfrac 1 2 \paren {r - x} }^{1/2}$

where:

$r$ is the modulus of $z$: $r = \sqrt {x^2 + y^2}$

the $\pm$ sign is taken to be the same as the sign of $y$.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Roots: $3.7.27$